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D₃×#535 #86329

order	384
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

Splittings

Fitting	C₃×#535 ⋊ C₂
Sylow	C₃ ⋊ C₂×#535

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₃
commutator, socle, cosocle	C₂²×C₃	C₂⁵
Fitting	C₃×#535	C₂
hypercenter	#535	D₃
hypocenter	C₃	C₂×#535
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ #535

1896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 48

A minimal faithful permutation representation — 5 generators on 15 points:

(1,2,3)(9,13)(12,14);(5,7)(10,15)(12,14);(8,10,11,15)(9,12,13,14);(4,5)(6,7)(8,9)(10,14)(11,13)(12,15);(2,3)

Conjugacy classes & character table

66 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×#1910, D₃×#1911, D₃×#1914, …
maximal normal subgroup of	D₃×#1910, D₃×#1911, D₃×#1914, …
maximal quotient of	D₃×#2309, D₃×#2311, D₃×#2317, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 18905);